3.11 \(\int (a+b x^2) (c+d x^2)^2 (e+f x^2) \, dx\)

Optimal. Leaf size=94 \[ \frac{1}{7} d x^7 (a d f+2 b c f+b d e)+\frac{1}{5} x^5 (a d (2 c f+d e)+b c (c f+2 d e))+\frac{1}{3} c x^3 (a c f+2 a d e+b c e)+a c^2 e x+\frac{1}{9} b d^2 f x^9 \]

[Out]

a*c^2*e*x + (c*(b*c*e + 2*a*d*e + a*c*f)*x^3)/3 + ((b*c*(2*d*e + c*f) + a*d*(d*e + 2*c*f))*x^5)/5 + (d*(b*d*e
+ 2*b*c*f + a*d*f)*x^7)/7 + (b*d^2*f*x^9)/9

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Rubi [A]  time = 0.0812315, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {521} \[ \frac{1}{7} d x^7 (a d f+2 b c f+b d e)+\frac{1}{5} x^5 (a d (2 c f+d e)+b c (c f+2 d e))+\frac{1}{3} c x^3 (a c f+2 a d e+b c e)+a c^2 e x+\frac{1}{9} b d^2 f x^9 \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)*(c + d*x^2)^2*(e + f*x^2),x]

[Out]

a*c^2*e*x + (c*(b*c*e + 2*a*d*e + a*c*f)*x^3)/3 + ((b*c*(2*d*e + c*f) + a*d*(d*e + 2*c*f))*x^5)/5 + (d*(b*d*e
+ 2*b*c*f + a*d*f)*x^7)/7 + (b*d^2*f*x^9)/9

Rule 521

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_))^(r_.), x_Symbol] :>
Int[ExpandIntegrand[(a + b*x^n)^p*(c + d*x^n)^q*(e + f*x^n)^r, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && I
GtQ[p, 0] && IGtQ[q, 0] && IGtQ[r, 0]

Rubi steps

\begin{align*} \int \left (a+b x^2\right ) \left (c+d x^2\right )^2 \left (e+f x^2\right ) \, dx &=\int \left (a c^2 e+c (b c e+2 a d e+a c f) x^2+(b c (2 d e+c f)+a d (d e+2 c f)) x^4+d (b d e+2 b c f+a d f) x^6+b d^2 f x^8\right ) \, dx\\ &=a c^2 e x+\frac{1}{3} c (b c e+2 a d e+a c f) x^3+\frac{1}{5} (b c (2 d e+c f)+a d (d e+2 c f)) x^5+\frac{1}{7} d (b d e+2 b c f+a d f) x^7+\frac{1}{9} b d^2 f x^9\\ \end{align*}

Mathematica [A]  time = 0.0288298, size = 96, normalized size = 1.02 \[ \frac{1}{5} x^5 \left (2 a c d f+a d^2 e+b c^2 f+2 b c d e\right )+\frac{1}{7} d x^7 (a d f+2 b c f+b d e)+\frac{1}{3} c x^3 (a c f+2 a d e+b c e)+a c^2 e x+\frac{1}{9} b d^2 f x^9 \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)*(c + d*x^2)^2*(e + f*x^2),x]

[Out]

a*c^2*e*x + (c*(b*c*e + 2*a*d*e + a*c*f)*x^3)/3 + ((2*b*c*d*e + a*d^2*e + b*c^2*f + 2*a*c*d*f)*x^5)/5 + (d*(b*
d*e + 2*b*c*f + a*d*f)*x^7)/7 + (b*d^2*f*x^9)/9

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Maple [A]  time = 0.002, size = 101, normalized size = 1.1 \begin{align*}{\frac{b{d}^{2}f{x}^{9}}{9}}+{\frac{ \left ( \left ( a{d}^{2}+2\,bcd \right ) f+b{d}^{2}e \right ){x}^{7}}{7}}+{\frac{ \left ( \left ( 2\,acd+b{c}^{2} \right ) f+ \left ( a{d}^{2}+2\,bcd \right ) e \right ){x}^{5}}{5}}+{\frac{ \left ( a{c}^{2}f+ \left ( 2\,acd+b{c}^{2} \right ) e \right ){x}^{3}}{3}}+a{c}^{2}ex \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)*(d*x^2+c)^2*(f*x^2+e),x)

[Out]

1/9*b*d^2*f*x^9+1/7*((a*d^2+2*b*c*d)*f+b*d^2*e)*x^7+1/5*((2*a*c*d+b*c^2)*f+(a*d^2+2*b*c*d)*e)*x^5+1/3*(a*c^2*f
+(2*a*c*d+b*c^2)*e)*x^3+a*c^2*e*x

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Maxima [A]  time = 1.01608, size = 135, normalized size = 1.44 \begin{align*} \frac{1}{9} \, b d^{2} f x^{9} + \frac{1}{7} \,{\left (b d^{2} e +{\left (2 \, b c d + a d^{2}\right )} f\right )} x^{7} + \frac{1}{5} \,{\left ({\left (2 \, b c d + a d^{2}\right )} e +{\left (b c^{2} + 2 \, a c d\right )} f\right )} x^{5} + a c^{2} e x + \frac{1}{3} \,{\left (a c^{2} f +{\left (b c^{2} + 2 \, a c d\right )} e\right )} x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)*(d*x^2+c)^2*(f*x^2+e),x, algorithm="maxima")

[Out]

1/9*b*d^2*f*x^9 + 1/7*(b*d^2*e + (2*b*c*d + a*d^2)*f)*x^7 + 1/5*((2*b*c*d + a*d^2)*e + (b*c^2 + 2*a*c*d)*f)*x^
5 + a*c^2*e*x + 1/3*(a*c^2*f + (b*c^2 + 2*a*c*d)*e)*x^3

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Fricas [A]  time = 1.26207, size = 282, normalized size = 3. \begin{align*} \frac{1}{9} x^{9} f d^{2} b + \frac{1}{7} x^{7} e d^{2} b + \frac{2}{7} x^{7} f d c b + \frac{1}{7} x^{7} f d^{2} a + \frac{2}{5} x^{5} e d c b + \frac{1}{5} x^{5} f c^{2} b + \frac{1}{5} x^{5} e d^{2} a + \frac{2}{5} x^{5} f d c a + \frac{1}{3} x^{3} e c^{2} b + \frac{2}{3} x^{3} e d c a + \frac{1}{3} x^{3} f c^{2} a + x e c^{2} a \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)*(d*x^2+c)^2*(f*x^2+e),x, algorithm="fricas")

[Out]

1/9*x^9*f*d^2*b + 1/7*x^7*e*d^2*b + 2/7*x^7*f*d*c*b + 1/7*x^7*f*d^2*a + 2/5*x^5*e*d*c*b + 1/5*x^5*f*c^2*b + 1/
5*x^5*e*d^2*a + 2/5*x^5*f*d*c*a + 1/3*x^3*e*c^2*b + 2/3*x^3*e*d*c*a + 1/3*x^3*f*c^2*a + x*e*c^2*a

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Sympy [A]  time = 0.073632, size = 121, normalized size = 1.29 \begin{align*} a c^{2} e x + \frac{b d^{2} f x^{9}}{9} + x^{7} \left (\frac{a d^{2} f}{7} + \frac{2 b c d f}{7} + \frac{b d^{2} e}{7}\right ) + x^{5} \left (\frac{2 a c d f}{5} + \frac{a d^{2} e}{5} + \frac{b c^{2} f}{5} + \frac{2 b c d e}{5}\right ) + x^{3} \left (\frac{a c^{2} f}{3} + \frac{2 a c d e}{3} + \frac{b c^{2} e}{3}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)*(d*x**2+c)**2*(f*x**2+e),x)

[Out]

a*c**2*e*x + b*d**2*f*x**9/9 + x**7*(a*d**2*f/7 + 2*b*c*d*f/7 + b*d**2*e/7) + x**5*(2*a*c*d*f/5 + a*d**2*e/5 +
b*c**2*f/5 + 2*b*c*d*e/5) + x**3*(a*c**2*f/3 + 2*a*c*d*e/3 + b*c**2*e/3)

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Giac [A]  time = 1.21616, size = 162, normalized size = 1.72 \begin{align*} \frac{1}{9} \, b d^{2} f x^{9} + \frac{2}{7} \, b c d f x^{7} + \frac{1}{7} \, a d^{2} f x^{7} + \frac{1}{7} \, b d^{2} x^{7} e + \frac{1}{5} \, b c^{2} f x^{5} + \frac{2}{5} \, a c d f x^{5} + \frac{2}{5} \, b c d x^{5} e + \frac{1}{5} \, a d^{2} x^{5} e + \frac{1}{3} \, a c^{2} f x^{3} + \frac{1}{3} \, b c^{2} x^{3} e + \frac{2}{3} \, a c d x^{3} e + a c^{2} x e \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)*(d*x^2+c)^2*(f*x^2+e),x, algorithm="giac")

[Out]

1/9*b*d^2*f*x^9 + 2/7*b*c*d*f*x^7 + 1/7*a*d^2*f*x^7 + 1/7*b*d^2*x^7*e + 1/5*b*c^2*f*x^5 + 2/5*a*c*d*f*x^5 + 2/
5*b*c*d*x^5*e + 1/5*a*d^2*x^5*e + 1/3*a*c^2*f*x^3 + 1/3*b*c^2*x^3*e + 2/3*a*c*d*x^3*e + a*c^2*x*e